Question

suppose the altitudes of a triangle are 10,12,15 find the semiperimeter

Answer 1

Let a, b, c are the sides of triangles.
we assume altitudes are 10, 12 and 15 on sides a, b , c respectively.
Let area of triangle is A
we know, area of ∆ = 1/2 × height × base

so, area of given triangle = 1/2 × 10 × a = 1/2 × 12 × b = 1/2 × 15 × c = A

a = A/5 , b = A/6 and C = 2A/15

now, S = (a + b + c)/2
= (A/5 + A/6 + 2A/15)/2
= (11A/30 + 2A/15)/2
= A/4 .......(i)

from Heron's formula,
area of given triangle = √{s(s - a)(s - b)(s - c)}
A = √{A/4(A/4 - A/5)(A/4 - A/6)(A/4 - 2A/15)}
A = √{A/4 × A/20 × A/12 × 7A/60}
A = A/2 × A/20 × 1/6 × √7
1 = √7A/240
A = 240/√7

so, S = A/4 = {240/√7}/4 = 60/√7

hence, answer is 60/√7

Answer 2

Here are the values as follow:

1/2 * a * 10 = 5a, 1/2 * b * 12 = 6b and 1/2 * c * 15 = 7.5c.  

Here, k is 5a and a = k/5, b = k/6 and c = 2k/15.  

Here is the heron’s formula which is sqrt{ s(s-a)(s-b)(s- c) }.

After applying the value, k = 240 /sqrt(7).

Since s is k/4.

Finally, we get s = = 22.68 unit.